The seamless development of mathematics and computation from a few clearly stated axioms and rules of inference in pure logic as embodied in an atomic/molecular medium. A gateway into the era of the molecule.

Immediately after fertilization the major role of the DNA machinery is to divide the cell and create an adult organism. The DNA and the Histone protein with it's associated tails are programmed for this task. The path drawn by the end carbon atom of the free end of the histone tail may give significant geometric temporal clues as to the function of the DNA. The tip serves as the Y in the equation Y=f(X), where X is some multidimensional vector. At any given point in time the tip of the histone tail will be in some given location in a 3 dimensional space. This may yield significant clues as to the geometrical properties related to the genetic machinery...as demonstrated by the similarities demonstrated by the above images of the cross section of an embryo and 2 possible strings created by a simulation of a histone tail. (Perry Moncznik, 2014)

The relative rotation velocities for the 20 spinning covalent bonds which generated this structure in the above image is: 0.008,0,0.008,2,4,0,2,4,0,2,4,0.008,0,0.008,2,4,0,2,4,0

The major difference between biomatics and other biological fields such as bioinformatics is that Biomatics is associated with design and construction of biological objects and machineries under mathematical principles.

The name Biomatics was coined in 1994 to differentiate a research field from Bioinformatics which is too computational and theoretical. Biomatics describes a field of biology that is not only theoretical but also has mechanical, designing, and synthetic aspects under clear top down principles.

Biological Mathematics or Principia BioMathematica

It is evident that some form of computation takes place in biological systems and indeed within single molecules. It thus follows that some form of mathematics occurs in these computations. Whether it be basic set theoretical concepts such as subsets, intersection, and union or more complex manipulations such as Fourier transforms of visual data.

The term “Biomatics” consists of a cross contraction of Biological Mathematics. The concept that is meant to be defined is the study of mathematics as it occurs in biological systems. This is in contrast to the concept of mathematical biology, which is the use of mathematics to describe or model biological systems.

A tentative definition might be: The seamless development of mathematics and computation from a few clearly stated axioms and rules of inference in pure logic as embodied in an atomic/molecular medium. Or, the study of "smart molecules".

Background

Gottfried Leibniz considered the following thesis in the late 17th century: (Some or all of) mathematics can be reduced to formal logic.

It is often described as a two-part thesis.
1. All mathematical truths can be translated into logical truths.
2. All mathematical proofs can be recast as logical proofs.
In other words, that all mathematical truths and proofs can be restated in the vocabulary of logic.

By the late 1800s Karl Weierstrass, Richard Dedekind and Georg Cantor had all developed methods for defining the irrationals in terms of the rationals. Giuseppe Peano had also gone on to develop a theory of the rationals based on his now famous axioms for the natural numbers. Thus, by Gottlob Frege's (predicate calculus) day (1848-1925) it was generally recognized that a large portion of mathematics could be derived from a relatively small set of primitive notions

In Bertrand Russell's words, it is the logicist's goal "to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms".

As a result, the question of whether mathematics can be reduced to logic, or whether it can be reduced only to set theory, remains open. However, in light of modern theories of evolution, fractal geometry, physics, chemistry and computer science, some concepts are now self-evident. Given that biological systems perform some sort of mathematics, and acceptance of evolution, it follows that these mathematical systems have evolved and therefore must have started from some initial state. Biomatics further raises at least the possibility that all of mathematics may be based on elemental algebraic structures as embodied in such molecules as the amino acids.

Intramolecular Computation

Consider an algebraic system embodied in a molecule consisting of N atoms. In the case where N = 3 we find the cube group (in terms of abstract algebra). (Note that N = 1 and N = 2 can represent groups as well).

Group theory (abstract algebra) is a well-developed branch of mathematics that provides many theorems and definitions. The key concept is that it describes, formally, a small (fundamental?) mathematical system consisting of a set and an operation on the members of that set. Could this then be nature’s way of evolving a system of mathematics and computation from a set of primitive notions? It seems it must inevitably be so, for ultimately what separates the different species, from viruses to man, is the complexity of the molecules that carry the blueprint for the ontogeny of the species.

As computer scientists, we seek and think in terms of information storage and processing. We seek to compare and contrast biological manifestations of computer science paradigms including: